*For non-mathematical meanings of "Integral", see integration (non-mathematical).*

In mathematics, the term "**integral**" has two unrelated meanings; one relating to integers, the other relating to **integral calculus**.

Table of contents |

2 Integral Calculus 3 Improper and Trigonometric Integrals 4 Means of Integration 5 Riemann and Lebesgue Integrals 6 Other integrals |

In calculus, the **integral**, of a function, is the size of the area bounded by the x-axis and the graph of a function, *f*(*x*); negative areas are possible. Integrals are calculated by **integration**, which is a so-called "accumulation process" (see below).

Let *f*(*x*) be a function of the interval [*a*,*b*] into the real numbers. For simplicity, assume that this function is non-negative (it takes no negative values.) The set *S*=*S _{f}*:={(

- disk integration
- list of integrals
- shell integration
- trigonometric integration

The antiderivative approach occurs when we seek to find a function *F*(*x*) whose derivative *F*(*x*) is some given function *f*(*x*). This approach is motivated by calculus, and is the main method used for calculating the area under the curve as described in the preceding paragraph, for functions given by formulae.

Functions which have antiderivatives are also Riemann integrable (and hence Lebesgue integrable.) The nonobvious theorem that states that the two approaches ("area under the curve" and "antiderivative") are in some sense the same is the fundamental theorem of calculus

*(And the relationships works in reverse; the Radon-Nikodym derivative can be pulled out of the measure machinery underlying Lebesgue integrals.)*

Lastly, a limit-taking step is taken to make the elementary functions approach *f* more and more closely, and an area is obtained for some functions *f*. The functions which we can integrate are said to be *integrable*. However, the differences begin here; the Riemann theory was simpler thus far, but its simplicity results in a more limited set of integrable functions than the Lebesgue theory. In addition, the interaction between limits and the integral are more difficult to describe in the Riemann setting.

- the Darboux integral, a variation of the Riemann integral
- the Denjoy integral, an extension of both the Riemann and Lebesgue integrals
- the Euler integral
- the Haar integral
- the Henstock-Kurzweil integral, an extension of both the Riemann and Lebesgue integrals (also called HK-integral)
- the Henstock-Kurzweil-Stieltjes integral (also called HK-Stieltjes integral)
- the Lebesgue-Stieltjes integral (also called Lebesgue-Radon integral)
- the Perron integral, which is equivalent to the restricted Denjoy integral
- the Stieltjes integral, an extension of the Riemann integral (also called Riemann-Stieltjes integral)

**See also**: Calculus, List of integrals